Alternative to Dzyaloshinskii-Moriya interaction for monolayer Fe$_3$GeTe$_2$ and other two-dimensional ferromagnets with trigonal prismatic symmetry.

We suggest a possible origin of noncollinear magnetic textures in ferromagnets (FMs) with the $D_{3h}$ point group symmetry. Suggested mechanism is different from the Dzyaloshinskii-Moriya interaction (DMI). Considered symmetry class is important because a large fraction of all single-layer intrinsic FMs should belong to it. In particular, so does a monolayer Fe$_3$GeTe$_2$. At the same time, DMI vanishes identically in materials described by this point group, in the continuous limit. We use symmetry analysis to derive all possible contributions to the free energy density that are of the fourth order with respect to the unit vector $\boldsymbol n$ of the local magnetization direction and linear with respect to spatial derivatives of $\boldsymbol n$. There are precisely seven such contributions. However, up to boundary terms, only two of them can be considered as independent. Moreover, for a two-dimensional system, one of these two necessarily vanishes! We investigate whether the remaining contribution can serve as a source of spin spirals and skyrmions recently observed in Fe$_3$GeTe$_2$. We also address a possible stabilization of bimerons by the same contribution.